Atlas › Modular forms › Weight 8 › Level 61 › 61.8.a › 61.8.a.a

Newform orbit 61.8.a.a

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q - 17q^{2} - 82q^{3} + 741q^{4} - 571q^{5} - 1295q^{6} - 355q^{7} - 3201q^{8} + 7314q^{9} - 971q^{10} - 18035q^{11} - 14009q^{12} - 4495q^{13} - 32184q^{14} - 25994q^{15} + 95341q^{16} - 29528q^{17} + 5142q^{18} - 56756q^{19} - 170873q^{20} - 121436q^{21} - 232781q^{22} - 340663q^{23} - 596965q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} + 99x^{15} - 16352x^{14} - 1622284x^{13} + 103699976x^{12} + 9826881320x^{11} - 345358898275x^{10} - 27570864062745x^{9} + 709302038473666x^{8} + 37250125907847464x^{7} - 902839201612347144x^{6} - 21223478014506106512x^{5} + 604278964808845391792x^{4} + 1942719037362214910896x^{3} - 149353388029218659045472x^{2} + 1293346987491953807417728x - 3423161070436268107790592$

Atkin–Lehner signs

$p$	$w_p$
61	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−17	−82	−571	−355	−18035	−4495	−29528	−56756	−340663	−670040	−497810	−109786	−219237	−1455730	176228	−1878614	−4956943	3631696	4562669	−9512338	4179019	−11499559	−4419572	−12894070	−77260834

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.291 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 61]: [1, 37] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(61, 8)[0]   # 61.8.a.a

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 61:8 1000